Quantum Mechanics => Operators

[jonnylane]

I'm doing some past papers for my QM finals and I've come across a question that is a bit strange. I'm not sure if it's as easy as it sounds.

X and P are one dimensional position and momentum operators, which take the explicit forms of x and -ihd/dx.

i) write down the explicit forms of X^2 and P^2

now then, is this just x^2 and hd^2/dx^2?

im ok on the next few bits, but:

iv) which, if any, of the two functions exp(ikx) and exp(-ax^2) are eigenfunctions of P^2?

my guess is that, since the eigenfunction of P is exp(ikx), its the other one (and the i has disapeared in the squaring process), but how can i prove this?


Im probably just being paranoid, but can someone verify these answers?

thanks

[ahrkron]

Originally posted by jonnylane
now then, is this just x^2 and hd^2/dx^2?

How did you obtain them? that's the important part. I think you're missing a sign on P^2.

iv) which, if any, of the two functions exp(ikx) and exp(-ax^2) are eigenfunctions of P^2?
how can i prove this?


Remember what an eigenfunction is. You need to plug in both functions on P^2, and see if you get a constant times the original function.